Numbers and Decimals

Movement of the Decimal Point

Adding and Subtracting Fractions and Decimals

Multiplying Decimals

Convert Fraction to Decimal Using Long Division

Convert Recurring Decimals to Fractions

Negative Numbers

Rational Numbers and their location on a Number Line

Adding and Subtracting Negative Numbers

Multiplying and Dividing Negative Numbers

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Convert Recurring Decimals to Fractions

Numbers and Decimals Convert Recurring Decimals to Fractions

Table Of Contents

🎬 Math Angel Video: Convert Recurring Decimals to Fractions

How to Convert Recurring Decimals to Fractions?

$Step-by-step guide converting 0.333 recurring decimal to fraction 1/3 by multiplying by 10, subtracting equations, and solving for x.$

⏩️ (0:01)

🛎 What Are Recurring Decimals?

Recurring Decimals are decimals that go on forever with a repeating pattern.

For example: $0.3333\ldots$ has the digit 3 repeating forever.

We can turn recurring decimals into fractions. Below is how:

🛎️ Example: Writing 0.333… into a fraction:

Let $x = 0.\dot{3}$ (the recurring decimal).
Multiply by 10: $$10x = 3.\dot{3}$$
Subtract the two equations. We do this to remove the recurring part:
$$
\begin{aligned}
10x−x &= 3.\dot{3}−0.\dot{3} \\[6pt]
9x &= 3
\end{aligned}
$$
Solve for $x$:
$$x = \tfrac{1}{3}$$

✅ So, $0.3333\ldots = \tfrac{1}{3}$.

Converting 0.4545... into a Fraction Step by Step

$Converting the recurring decimal 0.4545... into a fraction, multiplying both sides by 100, subtracting the equations, and solving for x = 5/11.$

⏩️ (0:46)

Let’s see another recurring decimal.

For example: $0.454545\ldots$ has the digits 45 repeating forever.

You can use the same method and steps to turn this recurring decimal into a fraction.

🛎️ Example: Writing 0.4545… into a fraction:

Let $x = 0.\dot{4}\dot{5}$ (the recurring decimal).
Multiply by 100: $$100x = 45.\dot{4}\dot{5}$$
Subtract the two equations. We do this to remove the recurring part: $$
\begin{aligned}
100x−x &= 45.\dot{4}\dot{5} −0.\dot{4}\dot{5} \\[6pt]
99x &= 45
\end{aligned}
$$
Solve for $x$:
$$x = \tfrac{45}{99} = \tfrac{5}{11}$$

✅ So, $0.454545\ldots = \tfrac{5}{11}$.

Converting 0.1666... into a Fraction Step by Step

$Converting the recurring decimal 0.1666... into a fraction, showing multiplication by powers of 10, subtraction of equations, and solving to get 1/6.$

⏩️ (1:34)

Sometimes the recurring part of a decimal does not start immediately after the decimal point.

For example: $0.1666\ldots$ has the digit 1 first, and then the 6 repeats forever.

We can still turn this recurring decimal into a fraction. Here’s how:

🛎️ Example: Writing 0.1666… into a fraction:

Let $x = 0.1\dot{6}$ (the recurring decimal).
Multiply by 10: $$10x = 1.\dot{6}$$
Multiply again by 10 (so that the recurring part lines up): $$100x = 16.\dot{6}$$
:Subtract the two equations. We do this to remove the recurring part: $$
\begin{aligned}
100x−10x &= 16.\dot{6}− 1.\dot{6} \\[6pt]
90x &= 15
\end{aligned}
$$
Solve for $x$:
$$x = \tfrac{15}{90} = \tfrac{1}{6}$$

✅ So, $0.1666\ldots = \tfrac{1}{6}$.

🍪 Quiz: Practice Converting Recurring Decimals to Fractions

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Convert Recurring Decimals to Fractions

1 / 6

Q: Convert $0.\overline{7}$ to a fraction.

$ \frac{7}{11} $

$ \frac{7}{10} $

$ \frac{7}{8} $

$ \frac{7}{9} $

Feedback: Not quite! Let $x = 0.\overline{7}$. Multiply both sides by 10: $10x = 7.\overline{7}$. Subtract $x = 0.\overline{7}$, leaving $9x = 7$. Solve for $x$: $x = \frac{7}{9}$.

Feedback: Well done! Let $x = 0.\overline{7}$. Multiply both sides by 10: $10x = 7.\overline{7}$. Subtract $x = 0.\overline{7}$, leaving $9x = 7$. Solve for $x$: $x = \frac{7}{9}$.

2 / 6

Q: Convert $0.\overline{12}$ to a fraction.

$ \frac{4}{25} $

$ \frac{13}{99} $

$ \frac{4}{33} $

$ \frac{6}{50} $

Feedback: Not quite! Let $x = 0.\overline{12}$. Multiply by 100 because the repeating part has two digits: $100x = 12.\overline{12}$. Subtract $x = 0.\overline{12}$, leaving $99x = 12$. Solve for $x$: $x = \frac{12}{99}$. Simplify the fraction: $x = \frac{4}{33}$.

Feedback: Well done! Let $x = 0.\overline{12}$. Multiply by 100 because the repeating part has two digits: $100x = 12.\overline{12}$. Subtract $x = 0.\overline{12}$, leaving $99x = 12$. Solve for $x$: $x = \frac{12}{99}$. Simplify the fraction: $x = \frac{4}{33}$.

3 / 6

Q: Convert $0.\overline{81}$ to a fraction.

$ \frac{27}{33} $

$ \frac{78}{99} $

$ \frac{10}{22} $

$ \frac{9}{11} $

Feedback: Not quite! Let $x = 0.\overline{81}$. Multiply both sides by 100 because the repeating part has two digits: $100x = 81.\overline{81}$. Subtract $x = 0.\overline{81}$, leaving $99x = 81$. Solve for $x$: $x = \frac{81}{99} = \frac{9}{11}$.

Feedback: Well done! Let $x = 0.\overline{81}$. Multiply both sides by 100 because the repeating part has two digits: $100x = 81.\overline{81}$. Subtract $x = 0.\overline{81}$, leaving $99x = 81$. Solve for $x$: $x = \frac{81}{99} = \frac{9}{11}$.

4 / 6

Q: Convert $0.1\overline{6}$ (0.1666 recurring) to a fraction.

$ \frac{1}{9} $

$ \frac{1}{8} $

$ \frac{1}{6} $

$ \frac{5}{30} $

Feedback: Not quite! Start with $x = 0.1\overline{6}$. Multiply by 10: $10x = 1.\overline{6}$. Then multiply by 100: $100x = 16.\overline{6}$. Subtract $10x = 1.\overline{6}$ to get $90x = 15$. Solve $x = \frac{15}{90} = \frac{1}{6}$.

Feedback: Well done! Start with $x = 0.1\overline{6}$. Multiply by 10: $10x = 1.\overline{6}$. Then multiply by 100: $100x = 16.\overline{6}$. Subtract $10x = 1.\overline{6}$ to get $90x = 15$. Solve $x = \frac{15}{90} = \frac{1}{6}$.

5 / 6

Q: Convert $0.\overline{123}$ to a fraction.

$ \frac{27}{297} $

$ \frac{123}{990} $

$ \frac{123}{1000} $

$ \frac{41}{333} $

Feedback: Not quite! Let $x = 0.\overline{123}$. Multiply by 1000 because the repeating part has three digits: $1000x = 123.\overline{123}$. Subtract $x = 0.\overline{123}$, leaving $999x = 123$. Solve for $x$: $x = \frac{123}{999}$. Simplify the fraction: $x = \frac{41}{333}$.

Feedback: Well done! Let $x = 0.\overline{123}$. Multiply by 1000 because the repeating part has three digits: $1000x = 123.\overline{123}$. Subtract $x = 0.\overline{123}$, leaving $999x = 123$. Solve for $x$: $x = \frac{123}{999}$. Simplify the fraction: $x = \frac{41}{333}$.

6 / 6

Q: Convert $0.2\overline{7}$ to a fraction.

$ \frac{22}{90} $

$ \frac{27}{99} $

$ \frac{28}{99} $

$ \frac{5}{18} $

Feedback: Not quite! Let $x = 0.2\overline{7}$. Multiply $x$ by 10: $10x = 2.\overline{7}$. Multiply the original equation by 100: $100x = 27.\overline{7}$. Subtract $10x = 2.\overline{7}$, leaving $90x = 25$. Solve for $x$: $x = \frac{25}{90} = \frac{5}{18}$.

Feedback: Well done! Let $x = 0.2\overline{7}$. Multiply $x$ by 10: $10x = 2.\overline{7}$. Multiply the original equation by 100: $100x = 27.\overline{7}$. Subtract $10x = 2.\overline{7}$, leaving $90x = 25$. Solve for $x$: $x = \frac{25}{90} = \frac{5}{18}$.

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